If `alpha, beta, gamma` are the roots of the equatiion `x^(3)-px^(2)+qx-r=0` find
`sumalpha^(2) beta`

Explanation

Since `alpha, beta, gamma` are the roots of `x^(3)-px^(2)+qx-r=0`
`:.sum alpha=p,sumalpha beta=q` and `alpha betagamma=r`
(i) `:'sum alpha. Sum alpha =p.p`
`implies(alpha+beta+gamma)(alpha+beta+gamma)=p^(2)`
`implies alpha^(2)+beta^(2)+gamma^(2)+2(alpha beta+beta gamma+gamma alpha)=p^(2)`
or `sum alpha^(2) +2 sum alpha beta=p^(2)`
or `sum alpha^(2)=p^(2)-2q`
(ii) `:'sum alpha . sum alpha beta=p.q`
`implies(alpha+beta+gamma).(alpha beta+beta gamma+gamma alpha)=pq`
`implies alpha^(2) beta+alpha beta gamma +alpha^(2)gamma +beta^(2) alpha+beta^(2)gamma+alpha beta gamma`
`+gamma^(2) beta+gamma^(2) alpha =pq` ltbr `implies (alpha^(2)beta+alpha^(2) alpha+beta^(2) gamma +beta^(2)gamma+gamma^(2) alpha+gamma^(2) beta)`
`+3 alpha beta gamma=pq`
or `sum alpha^(2) beta+3r=pq`
or `sum alpha^(2)beta=pq=3r`
(iii) `:'sum alpha^(2).sum alpha=(p^(2)-2q).p` [from result (i)]
`implies(alpha^(2)+beta^(2)=gamma^(2))(alpha+beta+gamma)=p^(3)-2pq`
`impliesalpha^(3)+beta^(3)+gamma^(3)+(alpha^(2)beta+alpha^(2) gamma+beta^(2) alpha+beta^(2)gamma`
`+gamma^(2) alpha+gamma^(2) beta)=p^(3)-2pq`
`impliessumalpha^(3)+sumalpha^(2) beta=p^(3)-2pq`
`impliessum alpha^(3)+pq-3r=p^(3)-2pq`[from result (ii) ]
or `sum alpha^(3)=p^(3)-3pq+3r`